# Abelian-extensible automorphism-invariant subgroup

From Groupprops

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Definition

Suppose is an abelian group and is a subgroup of . We say that is an **abelian-extensible automorphism-invariant subgroup** of if, for every abelian-extensible automorphism of , we have .

## Formalisms

### Function restriction expression

This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.

Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property

Function restriction expression | is a fully invariant subgroup of if ... | This means that full invariance is ... | Additional comments |
---|---|---|---|

abelian-extensible automorphism function | every abelian-extensible automorphism of sends every element of to within | the invariance property for abelian-extensible automorphisms | |

abelian-extensible automorphism endomorphism | every abelian-extensible automorphism of restricts to an endomorphism of | the endo-invariance property for abelian-extensible automorphisms; i.e., it is the invariance property for abelian-extensible automorphism, which is a property stronger than the property of being an endomorphism | |

abelian-extensible automorphism automorphism | every abelian-extensible automorphism of restricts to an automorphism of | the auto-invariance property for abelian-extensible automorphisms; i.e., it is the invariance property for abelian-extensible automorphism, which is a group-closed property of automorphisms |

## Relation with other properties

### Stronger properties

property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|

Characteristic subgroup of abelian group | Abelian-potentially characteristic subgroup|FULL LIST, MORE INFO | |||

Abelian-potentially characteristic subgroup | a characteristic subgroup in some bigger abelian group | abelian-potentially characteristic implies abelian-extensible automorphism-invariant | |FULL LIST, MORE INFO | |

Subgroup of finite abelian group | subgroup of finite abelian group | (via abelian-potentially characteristic; see finite abelian NPC theorem) | Abelian-potentially characteristic subgroup|FULL LIST, MORE INFO |

### Weaker properties

property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|

Subgroup of abelian group | subgroup of abelian group not implies abelian-extensible automorphism-invariant |